thm:discreteexpressive implies that the tangent vector \(\nabla _x \phi _{\epsilon }(x) \in T_x{\mathcal {M}}\)  converges in the Riemannian norm over \(T_x{\mathcal {M}}\)  to \(\nabla _x \phi (x) \in T_x{\mathcal {M}}\) . Furthermore, from the Hopf-Rinow Theorem \(\exp \)  is defined over all \(T_x{\mathcal {M}}\)  and it is continuous where it is defined [1]}. This shows the pointwise convergence.
